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I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that.

The (simplified) setting is the following. Let $p$ be a prime, and consider a homogeneous polynomial $f:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p$. I'm interested in sums of the form: $$ \sum_{x\in \mathcal{S}}\omega^{f(x)} $$ where $\omega=e^{2\pi i/p}$, and $\mathcal{S}\subseteq \mathbb{Z}_p^n$ is defined to be the set of solutions to the linear system of equations $\langle y_i , x \rangle=0$ for linearly independent $y_1,...,y_k\in \mathbb{Z}_p^n$ (assume $k<n$).

Any reference related to that would be useful (even possibly easier cases, e.g. $p=2, d=2$, would be very helpful).

Thanks!

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    $\begingroup$ I don't know any references specific to the case of exponential sums over subspaces, but the subspace is of course a vector space, and there exist many techniques for dealing with exponential sums over vector spaces, that can be potentially applied. For $p$ large (with respect to $d$) and $f$ good (in the sense of being a generic or random polynomial) there are geometric techniques as in Katz's notes on singular exponential sums. For $f$ structured there is the method of Weyl differencing. $\endgroup$
    – Will Sawin
    Commented Sep 11 at 15:48
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    $\begingroup$ For arbitrary $f$ there are ideas from combinatorics about structure vs. randomness e.g. results of Green and Tao on equidistribution of polynomials over finite fields and various followups to that. $\endgroup$
    – Will Sawin
    Commented Sep 11 at 15:51

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OP does not specify what they would like to know about such sums, so I will address the simplest thing that comes to mind: how can we evaluate them?

If $n=2$, then these are essentially the same thing as quadratic Gauss sums, hence, can be computed efficiently in polynomial time. If $n>2$, then without further constraints, the sums are #P-hard to evaluate. My favorite reference about such things is Cai, Chen, Lipton and Lu's paper "On Tractable Exponential Sums".

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  • $\begingroup$ I might note that my observations are mainly intended for the case $k=0$, i.e., when the subspace is the whole space $\mathbb{Z}_p^n$. So maybe I'm not understanding what the OP is really looking for. $\endgroup$
    – Eric S.
    Commented Sep 11 at 21:28

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